Oscillating Rim Hook Tableaux and Colored Matchings
William Y.C. Chen, Peter L. Guo

TL;DR
This paper establishes a novel combinatorial correspondence between oscillating m-rim hook tableaux and m-colored matchings, extending classical bijections and revealing new connections with Catalan numbers and Dyck path packings.
Contribution
It introduces a new bijection between oscillating m-rim hook tableaux and m-colored matchings, generalizing known results and linking to Dyck path packings.
Findings
Bijection between oscillating m-rim hook tableaux and m-colored matchings.
Special case for domino tableaux relating to noncrossing 2-colored matchings.
Dyck path packings counted by the product of two Catalan numbers.
Abstract
Motivated by the question of finding a type B analogue of the bijection between oscillating tableaux and matchings, we find a correspondence between oscillating m-rim hook tableaux and m-colored matchings, where m is a positive integer. An oscillating m-rim hook tableau is defined as a sequence of Young diagrams starting with the empty shape and ending with the empty shape such that is obtained from by adding an m-rim hook or by deleting an m-rim hook. Our bijection relies on the generalized Schensted algorithm due to White. An oscillating 2-rim hook tableau is also called an oscillating domino tableau. When we restrict our attention to two column oscillating domino tableaux of length 2n, we are led to a bijection between such tableaux and noncrossing 2-colored matchings on , which are counted by…
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · Algebraic structures and combinatorial models · Advanced Topics in Algebra
